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Theorem flimclslem 22007
Description: Lemma for flimcls 22008. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimcls.2 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
Assertion
Ref Expression
flimclslem ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆𝐹𝐴 ∈ (𝐽 fLim 𝐹)))

Proof of Theorem flimclslem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flimcls.2 . . 3 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
2 topontop 20937 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
323ad2ant1 1126 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
4 eqid 2770 . . . . . . . . 9 𝐽 = 𝐽
54neisspw 21131 . . . . . . . 8 (𝐽 ∈ Top → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝐽)
63, 5syl 17 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝐽)
7 toponuni 20938 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
873ad2ant1 1126 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋 = 𝐽)
98pweqd 4300 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝒫 𝑋 = 𝒫 𝐽)
106, 9sseqtr4d 3789 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝑋)
11 toponmax 20950 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
12 elpw2g 4955 . . . . . . . . . 10 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1311, 12syl 17 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1413biimpar 463 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
15143adant3 1125 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋)
1615snssd 4473 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ⊆ 𝒫 𝑋)
1710, 16unssd 3938 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋)
18 ssun2 3926 . . . . . 6 {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})
19113ad2ant1 1126 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋𝐽)
20 simp2 1130 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆𝑋)
2119, 20ssexd 4936 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V)
22 snnzg 4441 . . . . . . 7 (𝑆 ∈ V → {𝑆} ≠ ∅)
2321, 22syl 17 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ≠ ∅)
24 ssn0 4118 . . . . . 6 (({𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∧ {𝑆} ≠ ∅) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅)
2518, 23, 24sylancr 567 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅)
2620, 8sseqtrd 3788 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 𝐽)
27 simp3 1131 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ ((cls‘𝐽)‘𝑆))
284neindisj 21141 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ (𝐴 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥𝑆) ≠ ∅)
2928expr 444 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑥𝑆) ≠ ∅))
303, 26, 27, 29syl21anc 1474 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑥𝑆) ≠ ∅))
3130imp 393 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑥𝑆) ≠ ∅)
32 elsni 4331 . . . . . . . . . . 11 (𝑦 ∈ {𝑆} → 𝑦 = 𝑆)
3332ineq2d 3963 . . . . . . . . . 10 (𝑦 ∈ {𝑆} → (𝑥𝑦) = (𝑥𝑆))
3433neeq1d 3001 . . . . . . . . 9 (𝑦 ∈ {𝑆} → ((𝑥𝑦) ≠ ∅ ↔ (𝑥𝑆) ≠ ∅))
3531, 34syl5ibrcom 237 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑦 ∈ {𝑆} → (𝑥𝑦) ≠ ∅))
3635ralrimiv 3113 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → ∀𝑦 ∈ {𝑆} (𝑥𝑦) ≠ ∅)
3736ralrimiva 3114 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥𝑦) ≠ ∅)
38 simp1 1129 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ (TopOn‘𝑋))
394clsss3 21083 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
403, 26, 39syl2anc 565 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
4140, 27sseldd 3751 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 𝐽)
4241, 8eleqtrrd 2852 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴𝑋)
4342snssd 4473 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝐴} ⊆ 𝑋)
44 snnzg 4441 . . . . . . . . . 10 (𝐴 ∈ ((cls‘𝐽)‘𝑆) → {𝐴} ≠ ∅)
45443ad2ant3 1128 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝐴} ≠ ∅)
46 neifil 21903 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
4738, 43, 45, 46syl3anc 1475 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
48 filfbas 21871 . . . . . . . 8 (((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
4947, 48syl 17 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
50 ne0i 4067 . . . . . . . . . . 11 (𝐴 ∈ ((cls‘𝐽)‘𝑆) → ((cls‘𝐽)‘𝑆) ≠ ∅)
51503ad2ant3 1128 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
52 cls0 21104 . . . . . . . . . . 11 (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
533, 52syl 17 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘∅) = ∅)
5451, 53neeqtrrd 3016 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ≠ ((cls‘𝐽)‘∅))
55 fveq2 6332 . . . . . . . . . 10 (𝑆 = ∅ → ((cls‘𝐽)‘𝑆) = ((cls‘𝐽)‘∅))
5655necon3i 2974 . . . . . . . . 9 (((cls‘𝐽)‘𝑆) ≠ ((cls‘𝐽)‘∅) → 𝑆 ≠ ∅)
5754, 56syl 17 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ≠ ∅)
58 snfbas 21889 . . . . . . . 8 ((𝑆𝑋𝑆 ≠ ∅ ∧ 𝑋𝐽) → {𝑆} ∈ (fBas‘𝑋))
5920, 57, 19, 58syl3anc 1475 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ∈ (fBas‘𝑋))
60 fbunfip 21892 . . . . . . 7 ((((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋) ∧ {𝑆} ∈ (fBas‘𝑋)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥𝑦) ≠ ∅))
6149, 59, 60syl2anc 565 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥𝑦) ≠ ∅))
6237, 61mpbird 247 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
63 fsubbas 21890 . . . . . 6 (𝑋𝐽 → ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))))
6419, 63syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))))
6517, 25, 62, 64mpbir3and 1426 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋))
66 fgcl 21901 . . . 4 ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ∈ (Fil‘𝑋))
6765, 66syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ∈ (Fil‘𝑋))
681, 67syl5eqel 2853 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐹 ∈ (Fil‘𝑋))
69 fvex 6342 . . . . . 6 ((nei‘𝐽)‘{𝐴}) ∈ V
70 snex 5036 . . . . . 6 {𝑆} ∈ V
7169, 70unex 7102 . . . . 5 (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∈ V
72 ssfii 8480 . . . . 5 ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∈ V → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
7371, 72ax-mp 5 . . . 4 (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))
74 ssfg 21895 . . . . . 6 ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))))
7565, 74syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))))
7675, 1syl6sseqr 3799 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ 𝐹)
7773, 76syl5ss 3761 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝐹)
78 snssg 4448 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ↔ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
7921, 78syl 17 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ↔ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
8018, 79mpbiri 248 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))
8177, 80sseldd 3751 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆𝐹)
8277unssad 3939 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
83 elflim 21994 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
8438, 68, 83syl2anc 565 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
8542, 82, 84mpbir2and 684 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ (𝐽 fLim 𝐹))
8668, 81, 853jca 1121 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆𝐹𝐴 ∈ (𝐽 fLim 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1070   = wceq 1630  wcel 2144  wne 2942  wral 3060  Vcvv 3349  cun 3719  cin 3720  wss 3721  c0 4061  𝒫 cpw 4295  {csn 4314   cuni 4572  cfv 6031  (class class class)co 6792  ficfi 8471  fBascfbas 19948  filGencfg 19949  Topctop 20917  TopOnctopon 20934  clsccl 21042  neicnei 21121  Filcfil 21868   fLim cflim 21957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-en 8109  df-fin 8112  df-fi 8472  df-fbas 19957  df-fg 19958  df-top 20918  df-topon 20935  df-cld 21043  df-ntr 21044  df-cls 21045  df-nei 21122  df-fil 21869  df-flim 21962
This theorem is referenced by:  flimcls  22008
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